Nonradial solvability structure of super-diffusive nonlinear parabolic equations

Abstract:

We study the solvability of the Cauchy problem for the nonlinear parabolic equation \[ \frac {\partial u}{\partial t} = \mbox {div} (u^{m-1}\nabla u)\] when $m < 0$ in $\textbf {R}^2$, with $u(x,0)= f(x)$ a given nonnegative function. It is known from earlier works of the authors that the asymptotic radial growth $r^{-2/1-m}$, $r=|x|$ for the spherical averages of $f(x)$ is critical for local solvability, in particular ensuring it if $f$ is radially symmetric. We show that if the initial data $f(x)$ behaves in polar coordinates like $r^{-2/1-m} g(\theta )$, for large $r= |x|$ with $g$ nonnegative and $2\pi$-periodic, then the following holds: If $g$ vanishes on some interval of length $l^* = \frac {(m-1)\pi }{2m} >0$, then there is no local solution of the initial value problem. On the other hand, if such an interval does not exist, then the initial value problem is locally solvable and the time of existence can be estimated explicitly.